f[x_] := Abs[Differences[x]]~Join~{Abs[First[x] - Last[x]]};
points = (RegionPlot3D[
       x^2 + y^2 + z^2 < 1, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] // 
      InputForm) /. 
    Graphics3D[GraphicsComplex[points_, ___], _] :> points // First;
{u, v, w} = Orthogonalize[NullSpace[{RandomReal[1, 4]}]];
Table[ListPointPlot3D[
  Nest[({u, v, w}.f[u #[[1]] + v #[[2]] + w #[[3]]]) & /@ # &, points,
    l], ImageSize -> {300, 300}, AspectRatio -> 1, 
  PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}], {l, 0, 8}]